Timeline for Symmetric monoidal category with trivial switch morphisms
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 11 at 12:03 | comment | added | Martin Brandenburg | Ah yeah, up to equivalence. That is OK for me. My comment was about the (wrong) statement that if a symmetric monoidal category has trivial self-braidings, it is commutative. | |
| Mar 14 at 11:25 | comment | added | Léo S. | EDIT: according to an article of Kim (cited by the nLab), it is indeed true that, under some assumptions on size, a symmetric monoidal category is symmetric monoidal equivalent to a commutative monoidal category if and only if all its self-braidings are identities. | |
| Mar 14 at 11:03 | comment | added | Léo S. | @MartinBrandenburg Why doesn't the argument presented by Simon Henry work? I had the same intuition: forcing self-symmetries to be identities implies coherence (there is a unique way to permute a given tensor product), and hence should lead to a strictification theorem where permutations are identities. | |
| Jan 30, 2020 at 18:50 | comment | added | Simon Henry | @MartinBrandenburg : You know the topic better than I do, but I was under the impression that an appropriate modification of the proof of the strictification theorem was proving this: the condition that the switch map $x \otimes x \rightarrow x \otimes x$ is the identity shows that more general permutation of identical objects in a tensor product all acts as the identity. So one can consider the category MC whose objects are multi-set of objects of C, and morphisms are map between their tensor product (for any ordering) ? $\otimes$ is union and is strictly commutative. | |
| Jan 30, 2020 at 18:04 | comment | added | Martin Brandenburg | @SimonHenry Really? | |
| Jul 21, 2019 at 16:03 | history | bounty ended | CommunityBot | ||
| Jul 19, 2019 at 22:51 | vote | accept | Captain Lama | ||
| Jul 14, 2019 at 13:48 | comment | added | Simon Henry | They are indeed equivalent in this way. | |
| Jul 14, 2019 at 9:31 | comment | added | Captain Lama | Thanks for the reference. This being said, this seems quite a lot stronger than what I'm thinking about (the example that prompted my question does not satisfy such a strong property). Is it possible that any monoidal category satisfying the property in my question is monoidally equivalent to one as in your answer ? | |
| Jul 13, 2019 at 20:11 | history | answered | Noam Zeilberger | CC BY-SA 4.0 |